UBC Theses and Dissertations
Surface codes, the 2D classical Ising model, and non-interacting fermions Goff, Leonard Thomas
In this thesis, we consider the task of simulating measurement-based quantum computation (MBQC) on surface code states: the generalization of Kitaev's toric code to graphs embedded on a surface of higher genus. We define a family of higher genus graphs and a simple ordering of single qubit measurements, and find that simulating MBQC on any of the associated surface code states is equivalent to evaluating the inner-product between a product state and a surface code state on another graph. We further find that such an inner-product can always be written as a sum of one or more 2D classical Ising model partition functions, with appropriate couplings. For certain higher genus square lattices, we develop a means to evaluate this partition function in a number of steps that scales polynomially in the number of qubits, but exponentially in the genus of the embedded graph. The method makes use of the transfer matrix formalism for the Ising partition function, and a subsequent mapping to fermion operators. We synthesize these results to relate the simulation of MBQC on certain surface code states to a system of fermions interacting with the encoded qubits of the surface code. We identify a family of states in the code space of the surface code on our higher genus graphs for which MBQC can be simulated efficiently in all parameters, including the genus of the embedded graph. Finally, we identify two connections between the complexity of this task and entanglement.
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